#3659473

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,5,7,8)(10,11)(12,16)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,40,36,39);(10,11)(14,15)(17,19,20,23)(18,21,22,24)(26,30)(31,38)(32,35)(39,40);(9,10,13,11)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,23)(22,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,3,4,7)(2,5,6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=fh,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3566669);